Flat quads (or, why is my mesh so funky?)
As you already suspect, in three-dimensional digital graphics computer basically only know how to draw two things: Lines and Triangles
There are a few other primitives types (point clouds, volumes, voxels etc.) but for the sake of simplicity lets focus on the former two.
Historically speaking, in the first 3D modelling software you could only model with triangle based geometry, because it's the only unequivocally definable 3D structure with three points. Even Blender itself a few years ago as of versions pre 2.63 (when BMesh geometry engine was introduces) was still only able to define faces with 3 or 4 sides, not more.
As technology evolved it started being practical to define faces with more than three sides.
A quad, as you correctly stated is not unequivocally defined by those four points, it is not however mathematically undefined. Since computers only know how to draw triangles, in the background the software is trying to silently triangulate the shapes for you as best as it can, as you can see bellow.

Simple shapes are trivial, but more complex shapes (where supported by the software) are even more difficult to triangulate, and may often generate artifacts from unexpected or uncontrollable tessellation. That is one of the reasons why you will often hear people say you should strive to have a quad dominant topology.
Blender doesn't explicitly allow tessellation control, but other software or mesh engines some times have tools that allow controlling how a shape is triangulated (as is the case of 3DSMax 3D Polys, but not meshes).

TL;DR
Should you be worried by this? No, not in most cases.
This is normal behavior and non-planar shapes are totally expected (if not the norm) everywhere in your mesh. Modelling with planar faces everywhere would be extremely hard, or certain shapes would be downright impossible to achieve.
Some extreme cases can some times cause issues and should be avoided though, like concave planar faces, or very high distortion quads, but most others occurrences should be fine.
There are builtin tools in Blender to help with these, like the Mesh &gt; Cleanup &gt; Make Planar Faces, to help detect where issues might occur.
What you seem to be missing is the concept of subdivision. Except for hard-edge models most organic shapes are frequently modeled using a Subdivide Modifier, that will take care of automatically smoothing your surfaces. It will soften your geometry angles, while retaining a simple enough base mesh to work with, so that quads appear to have a smoother appearance.